Some remarks on the linearized operator about the radial solution for the Ginzburg–Landau equation Original Research Article

We consider the linearized operators, denoted View the MathML source, of the Ginzburg–Landau operator Δu+u(1−|u|2) in R2, about the radial solutions ud,1(x)=fd(r)eidθ, for all d⩾1. We state the correspondence between the real vector space of the bounded solutions of the equation View the MathML source and the eigenvalues of the linearized operators of the equations Δu+1/ε2u(1−|u|2)=0, in B(0,1), about the radial solutions ud,ε(x)=fd(r/ε)eidθ, that tend to 0 as ε tends to 0.